Question

problem 2 complete the flowchart proof to show that ∠t ≅ ∠q if \\(\overline{st} \parallel \overline{qp}\\) and \\(\overline{tp} \parallel \overline{sq}\\). \\(\angle pst\\) and \\(\angle spq\\) are alternate interior angles \\(\angle pst \cong \angle spq\\) given \\(\overline{tp} \parallel \overline{sq}\\) definition of alternate interior angles alternate interior angles theorem \\(\underline{\quad\quad}\\) triangle congruence theorem

Explanation:

Step1: Identify the first given condition

The first blank is related to the condition that leads to the statement about alternate - interior angles. Since \( \angle PST\) and \( \angle SPQ\) are alternate - interior angles, the first condition is \( \overline{ST}\parallel\overline{QP}\) (Given).

Step2: For \( \overline{TP}\parallel\overline{SQ}\)

When \( \overline{TP}\parallel\overline{SQ}\), \( \angle TPS\) and \( \angle QSP\) are alternate - interior angles (by the definition of alternate - interior angles). Then, by the alternate - interior angles theorem, \( \angle TPS\cong\angle QSP\).

Step3: Congruent triangles

We have two pairs of congruent angles (\( \angle PST\cong\angle SPQ\) and \( \angle TPS\cong\angle QSP\)) and the common side \( \overline{SP}\). So, by the ASA (Angle - Side - Angle) triangle congruence theorem, \( \triangle STP\cong\triangle PQS\).

Step4: Corresponding parts of congruent triangles

Since \( \triangle STP\cong\triangle PQS\), by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, \( \angle T\cong\angle Q\).

Answer:

1. \( \overline{ST}\parallel\overline{QP}\)
2. \( \angle TPS\) and \( \angle QSP\) are alternate - interior angles
3. \( \angle TPS\cong\angle QSP\)
4. ASA
5. \( \triangle STP\cong\triangle PQS\)
6. CPCTC, \( \angle T\cong\angle Q\)